How to obtain the following topology?
Unless I've severely misunderstood what you're trying to achieve, I still think a torus with Minor Segments set to 4 is the easiest and quickest way.

ShiftA followed by M followed by T or Add -> Mesh -> Torus to add the torus.



After adding it and changing Minor Segments to 4, just scale along Z, S followed by Z, then type in the scale or drag the mouse, and finish with Enter or LMB.